Deflating Quadratic Matrix Polynomials

Abstract

In this thesis we consider algorithms for solving the quadratic eigenvalue problem,
(lambda^2*A_2 + lambda*A_1 + A_0)x=0
when the leading or trailing
coefficient matrices are singular. In a finite element discretization this corresponds to the mass or stiffness matrices
being singular
and reflects modes of vibration (or eigenvalues) at zero or ``infinity''. We are interested in deflation procedures
that enable us to utilize knowledge of the presence of these (or any) eigenvalues to reduce the overall cost in
computing the remaining eigenvalues and eigenvectors of interest.
We first give an introduction to the quadratic eigenvalue problem and explain how it can be solved by a process called linearization.
We present two types of algorithms, firstly a modification of an algorithm published by
Kublanovskaya, Mikhailov, and Khazanov in the 1970s that has recently been translated into English.
Using these ideas we present algorithms that are able to reduce the size of the problem by ``deflating''
infinite and zero eigenvalues that arise when the mass or stiffness matrix (or both) are singular.
Secondly we look at methods that deflate zero and infinite eigenvalues by the use of Householder reflectors;
this requires a basis for the null space of the mass or stiffness matrix (or both), so we also summarize various decompositions
that can be used to give this information.
We consider different applications that yield a quadratic eigenvalue problem
with singular leading and trailing coefficients and after testing the implementations of the algorithms
on some of these problems we comment on their stability.